We may assume \(\Vert \Omega \Vert _{L^{\infty }(\mathbb {S}^{d-1})}=1\)  and claim that for each \(n\)  \((1\le n\le d)\) ,
the inequality
\(|\Lambda _{n,\mathcal {Q}}(f_1,\,f_2)|\lesssim r^{\prime }|Q|\Vert f_2\Vert _{\mathcal {Y}_{r}(\mathcal {Q})}\) 
holds provided that \({\rm supp} f_1\subset Q\)  and \(\langle |f_1|\rangle _{Q}\le 1\) .
In fact, applying the Calderón-Zygmund decomposition of \(f_1\)  at level 1
with respect to \(\mathcal {Q}\) , we may decompose \(f_1\)  as \(f_1=b_1+g_1\) ,
with the properties that
\(\Vert g_1\Vert _{L^{\infty }(\mathbb {R}^d)} \lesssim 1, \qquad {\rm supp}\,g_1\subset Q, \qquad \Vert b_1\Vert _{\dot{\mathcal {X}}_{1}(\mathcal {Q})}\lesssim 1.\) 
An application of [1]} may lead to that,
\(|\Lambda _{n,\mathcal {Q}}(g_1,\,f_2)|&\lesssim &r^{\prime }\Vert g_1\Vert _{L^{r^{\prime }}(\mathbb {R}^{d})} \Vert f_2\Vert _{L^{r}(\mathbb {R}^{d})}\nonumber \lesssim r^{\prime }|Q| \Vert f_2\Vert _{\mathcal {Y}_{r}(\mathcal {Q})}.\nonumber \) 
On the other hand, by [1]} and the fact that \(\Vert b_1\Vert _{\dot{\mathcal {X}}_{1}(\mathcal {Q})}\lesssim 1\) , we obtain
\(| \Lambda _{n, \mathcal {Q}}(b_1,\,f_2)|\lesssim r^{\prime }|Q| \Vert {f_2}\Vert _{\mathcal {Y}_{r}(\mathcal {Q})}, \) 

which verifies this claim.
Now we return to the proof of (REF ). By homogeneity, we may assume \(\Vert b\Vert _{\dot{\mathcal {X}}_{L\log L}(\mathcal {Q})}=1\) , which means that for each \(\mathbb {L}\in \mathcal {Q}\) ,
\(\Vert b_\mathbb {L}\Vert _{L\log L,\,L}\le 1.\) 
Observe that \({\rm supp}\,\sum \limits _{\mathbb {L}\in \mathcal {Q} }b_\mathbb {L}\partial _nA_{\mathbb {L}}\subset Q\) , and by (REF ),
\(\Big \Vert \sum \limits _{\mathbb {L}\in \mathcal {Q} }b_\mathbb {L}\partial _nA_{\mathbb {L}}\Big \Vert _{ L^{1}(\mathbb {R}^{d})}\lesssim \sum \limits _{\mathbb {L}\in \mathcal {Q} }|\mathbb {L}|\Vert b_\mathbb {L}\Vert _{ {L \log L, L}}\lesssim |Q|.\) 
Our claim now leads to
\(\Big |\Lambda _{n,\mathcal {Q}}(\sum \limits _{\mathbb {L}\in \mathcal {Q} }\partial _nA_{\mathbb {L}}b_\mathbb {L},\,f_2) \Big | \le r^{\prime }|Q|\Vert {f_2}\Vert _{\mathcal {Y}_{r}(\mathcal {Q})},\) 
which completes the proof of Lemma REF .
